3.1853 \(\int (a+b x)^2 (c+d x)^n \, dx\)
Optimal. Leaf size=78 \[ \frac{(b c-a d)^2 (c+d x)^{n+1}}{d^3 (n+1)}-\frac{2 b (b c-a d) (c+d x)^{n+2}}{d^3 (n+2)}+\frac{b^2 (c+d x)^{n+3}}{d^3 (n+3)} \]
[Out]
((b*c - a*d)^2*(c + d*x)^(1 + n))/(d^3*(1 + n)) - (2*b*(b*c - a*d)*(c + d*x)^(2 + n))/(d^3*(2 + n)) + (b^2*(c
+ d*x)^(3 + n))/(d^3*(3 + n))
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Rubi [A] time = 0.0320649, antiderivative size = 78, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used =
{43} \[ \frac{(b c-a d)^2 (c+d x)^{n+1}}{d^3 (n+1)}-\frac{2 b (b c-a d) (c+d x)^{n+2}}{d^3 (n+2)}+\frac{b^2 (c+d x)^{n+3}}{d^3 (n+3)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2*(c + d*x)^n,x]
[Out]
((b*c - a*d)^2*(c + d*x)^(1 + n))/(d^3*(1 + n)) - (2*b*(b*c - a*d)*(c + d*x)^(2 + n))/(d^3*(2 + n)) + (b^2*(c
+ d*x)^(3 + n))/(d^3*(3 + n))
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int (a+b x)^2 (c+d x)^n \, dx &=\int \left (\frac{(-b c+a d)^2 (c+d x)^n}{d^2}-\frac{2 b (b c-a d) (c+d x)^{1+n}}{d^2}+\frac{b^2 (c+d x)^{2+n}}{d^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (c+d x)^{1+n}}{d^3 (1+n)}-\frac{2 b (b c-a d) (c+d x)^{2+n}}{d^3 (2+n)}+\frac{b^2 (c+d x)^{3+n}}{d^3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0677709, size = 67, normalized size = 0.86 \[ \frac{(c+d x)^{n+1} \left (-\frac{2 b (c+d x) (b c-a d)}{n+2}+\frac{(b c-a d)^2}{n+1}+\frac{b^2 (c+d x)^2}{n+3}\right )}{d^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2*(c + d*x)^n,x]
[Out]
((c + d*x)^(1 + n)*((b*c - a*d)^2/(1 + n) - (2*b*(b*c - a*d)*(c + d*x))/(2 + n) + (b^2*(c + d*x)^2)/(3 + n)))/
d^3
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Maple [B] time = 0.007, size = 159, normalized size = 2. \begin{align*}{\frac{ \left ( dx+c \right ) ^{1+n} \left ({b}^{2}{d}^{2}{n}^{2}{x}^{2}+2\,ab{d}^{2}{n}^{2}x+3\,{b}^{2}{d}^{2}n{x}^{2}+{a}^{2}{d}^{2}{n}^{2}+8\,ab{d}^{2}nx-2\,{b}^{2}cdnx+2\,{b}^{2}{d}^{2}{x}^{2}+5\,{a}^{2}{d}^{2}n-2\,abcdn+6\,ab{d}^{2}x-2\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}-6\,abcd+2\,{b}^{2}{c}^{2} \right ) }{{d}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2*(d*x+c)^n,x)
[Out]
(d*x+c)^(1+n)*(b^2*d^2*n^2*x^2+2*a*b*d^2*n^2*x+3*b^2*d^2*n*x^2+a^2*d^2*n^2+8*a*b*d^2*n*x-2*b^2*c*d*n*x+2*b^2*d
^2*x^2+5*a^2*d^2*n-2*a*b*c*d*n+6*a*b*d^2*x-2*b^2*c*d*x+6*a^2*d^2-6*a*b*c*d+2*b^2*c^2)/d^3/(n^3+6*n^2+11*n+6)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(d*x+c)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.93191, size = 478, normalized size = 6.13 \begin{align*} \frac{{\left (a^{2} c d^{2} n^{2} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} +{\left (b^{2} d^{3} n^{2} + 3 \, b^{2} d^{3} n + 2 \, b^{2} d^{3}\right )} x^{3} +{\left (6 \, a b d^{3} +{\left (b^{2} c d^{2} + 2 \, a b d^{3}\right )} n^{2} +{\left (b^{2} c d^{2} + 8 \, a b d^{3}\right )} n\right )} x^{2} -{\left (2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} n +{\left (6 \, a^{2} d^{3} +{\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} n^{2} -{\left (2 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(d*x+c)^n,x, algorithm="fricas")
[Out]
(a^2*c*d^2*n^2 + 2*b^2*c^3 - 6*a*b*c^2*d + 6*a^2*c*d^2 + (b^2*d^3*n^2 + 3*b^2*d^3*n + 2*b^2*d^3)*x^3 + (6*a*b*
d^3 + (b^2*c*d^2 + 2*a*b*d^3)*n^2 + (b^2*c*d^2 + 8*a*b*d^3)*n)*x^2 - (2*a*b*c^2*d - 5*a^2*c*d^2)*n + (6*a^2*d^
3 + (2*a*b*c*d^2 + a^2*d^3)*n^2 - (2*b^2*c^2*d - 6*a*b*c*d^2 - 5*a^2*d^3)*n)*x)*(d*x + c)^n/(d^3*n^3 + 6*d^3*n
^2 + 11*d^3*n + 6*d^3)
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Sympy [A] time = 1.99613, size = 1504, normalized size = 19.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**2*(d*x+c)**n,x)
[Out]
Piecewise((c**n*(a**2*x + a*b*x**2 + b**2*x**3/3), Eq(d, 0)), (-a**2*c*d**2/(2*c**3*d**3 + 4*c**2*d**4*x + 2*c
*d**5*x**2) + 2*a*b*d**3*x**2/(2*c**3*d**3 + 4*c**2*d**4*x + 2*c*d**5*x**2) + 2*b**2*c**3*log(c/d + x)/(2*c**3
*d**3 + 4*c**2*d**4*x + 2*c*d**5*x**2) + b**2*c**3/(2*c**3*d**3 + 4*c**2*d**4*x + 2*c*d**5*x**2) + 4*b**2*c**2
*d*x*log(c/d + x)/(2*c**3*d**3 + 4*c**2*d**4*x + 2*c*d**5*x**2) + 2*b**2*c*d**2*x**2*log(c/d + x)/(2*c**3*d**3
+ 4*c**2*d**4*x + 2*c*d**5*x**2) - 2*b**2*c*d**2*x**2/(2*c**3*d**3 + 4*c**2*d**4*x + 2*c*d**5*x**2), Eq(n, -3
)), (-a**2*d**2/(c*d**3 + d**4*x) + 2*a*b*c*d*log(c/d + x)/(c*d**3 + d**4*x) + 2*a*b*c*d/(c*d**3 + d**4*x) + 2
*a*b*d**2*x*log(c/d + x)/(c*d**3 + d**4*x) - 2*b**2*c**2*log(c/d + x)/(c*d**3 + d**4*x) - 2*b**2*c**2/(c*d**3
+ d**4*x) - 2*b**2*c*d*x*log(c/d + x)/(c*d**3 + d**4*x) + b**2*d**2*x**2/(c*d**3 + d**4*x), Eq(n, -2)), (a**2*
log(c/d + x)/d - 2*a*b*c*log(c/d + x)/d**2 + 2*a*b*x/d + b**2*c**2*log(c/d + x)/d**3 - b**2*c*x/d**2 + b**2*x*
*2/(2*d), Eq(n, -1)), (a**2*c*d**2*n**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 5*a**2*c
*d**2*n*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a**2*c*d**2*(c + d*x)**n/(d**3*n**3 +
6*d**3*n**2 + 11*d**3*n + 6*d**3) + a**2*d**3*n**2*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**
3) + 5*a**2*d**3*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a**2*d**3*x*(c + d*x)**n/
(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) - 2*a*b*c**2*d*n*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**
3*n + 6*d**3) - 6*a*b*c**2*d*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*a*b*c*d**2*n**2*x
*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a*b*c*d**2*n*x*(c + d*x)**n/(d**3*n**3 + 6*d*
*3*n**2 + 11*d**3*n + 6*d**3) + 2*a*b*d**3*n**2*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**
3) + 8*a*b*d**3*n*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 6*a*b*d**3*x**2*(c + d*x)
**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*b**2*c**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d*
*3*n + 6*d**3) - 2*b**2*c**2*d*n*x*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*c*d**2*n
**2*x**2*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*c*d**2*n*x**2*(c + d*x)**n/(d**3*n
**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + b**2*d**3*n**2*x**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*
n + 6*d**3) + 3*b**2*d**3*n*x**3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3) + 2*b**2*d**3*x**
3*(c + d*x)**n/(d**3*n**3 + 6*d**3*n**2 + 11*d**3*n + 6*d**3), True))
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Giac [B] time = 1.07337, size = 520, normalized size = 6.67 \begin{align*} \frac{{\left (d x + c\right )}^{n} b^{2} d^{3} n^{2} x^{3} +{\left (d x + c\right )}^{n} b^{2} c d^{2} n^{2} x^{2} + 2 \,{\left (d x + c\right )}^{n} a b d^{3} n^{2} x^{2} + 3 \,{\left (d x + c\right )}^{n} b^{2} d^{3} n x^{3} + 2 \,{\left (d x + c\right )}^{n} a b c d^{2} n^{2} x +{\left (d x + c\right )}^{n} a^{2} d^{3} n^{2} x +{\left (d x + c\right )}^{n} b^{2} c d^{2} n x^{2} + 8 \,{\left (d x + c\right )}^{n} a b d^{3} n x^{2} + 2 \,{\left (d x + c\right )}^{n} b^{2} d^{3} x^{3} +{\left (d x + c\right )}^{n} a^{2} c d^{2} n^{2} - 2 \,{\left (d x + c\right )}^{n} b^{2} c^{2} d n x + 6 \,{\left (d x + c\right )}^{n} a b c d^{2} n x + 5 \,{\left (d x + c\right )}^{n} a^{2} d^{3} n x + 6 \,{\left (d x + c\right )}^{n} a b d^{3} x^{2} - 2 \,{\left (d x + c\right )}^{n} a b c^{2} d n + 5 \,{\left (d x + c\right )}^{n} a^{2} c d^{2} n + 6 \,{\left (d x + c\right )}^{n} a^{2} d^{3} x + 2 \,{\left (d x + c\right )}^{n} b^{2} c^{3} - 6 \,{\left (d x + c\right )}^{n} a b c^{2} d + 6 \,{\left (d x + c\right )}^{n} a^{2} c d^{2}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(d*x+c)^n,x, algorithm="giac")
[Out]
((d*x + c)^n*b^2*d^3*n^2*x^3 + (d*x + c)^n*b^2*c*d^2*n^2*x^2 + 2*(d*x + c)^n*a*b*d^3*n^2*x^2 + 3*(d*x + c)^n*b
^2*d^3*n*x^3 + 2*(d*x + c)^n*a*b*c*d^2*n^2*x + (d*x + c)^n*a^2*d^3*n^2*x + (d*x + c)^n*b^2*c*d^2*n*x^2 + 8*(d*
x + c)^n*a*b*d^3*n*x^2 + 2*(d*x + c)^n*b^2*d^3*x^3 + (d*x + c)^n*a^2*c*d^2*n^2 - 2*(d*x + c)^n*b^2*c^2*d*n*x +
6*(d*x + c)^n*a*b*c*d^2*n*x + 5*(d*x + c)^n*a^2*d^3*n*x + 6*(d*x + c)^n*a*b*d^3*x^2 - 2*(d*x + c)^n*a*b*c^2*d
*n + 5*(d*x + c)^n*a^2*c*d^2*n + 6*(d*x + c)^n*a^2*d^3*x + 2*(d*x + c)^n*b^2*c^3 - 6*(d*x + c)^n*a*b*c^2*d + 6
*(d*x + c)^n*a^2*c*d^2)/(d^3*n^3 + 6*d^3*n^2 + 11*d^3*n + 6*d^3)